Lecture 7. Order Out of Chaos

Lecture 7 Order Out of Chaos

Lyapunov Exponents: Recall from Last Time The Lyapunov exponent for maps is: < 0if f 0 (x ) < 1 =ln f 0 (x ) = > 0if f 0 (x ) > 1

Lyapunov Exponents: Numerical Estimation Lyapunov exponent: =ln f 0 (x ) Rate of divergence over time: f 0 (x) = dx t+1 dx t

Lyapunov Exponents: Numerical Estimation Lyapunov exponent: =ln f 0 (x ) Rate of divergence over time: f 0 (x) = dx t+1 dx t Assumption: Lyapunov exponent is the same everywhere in the basin of attraction.

Lyapunov Exponents: Maximum 1 =lim max t 1 t ln f t(x) dx This is really the averaged sum of the Lyapunov exponents,.

Lyapunov Exponents: Maximum : Short term behaviour max: Long term behaviour

Lyapunov Exponents: Numerical Estimation Since maps depend on the previous step, we can show that: Mean rate of divergence: max =lim n 1 1 t ln f t(x) dx = 1 t max t max X ln dx t+1 dx t=t t 0

Lyapunov Exponents: Numerical Estimation This is our approximation of max 1 t max tx max ln dx t+1 dx t t=t 0

Lyapunov Exponents: Numerical Estimation We want to look at the behaviour near x. How can we know x t is near x?

Lyapunov Exponents: Numerical Estimation We want to look at the behaviour near x. How can we know x t is near x? We iterate the map long enough that x t crosses the separatrix that defines the long term behaviour.

Lyapunov Exponents: Numerical Estimation This is our approximation of : max max = 1 t max t max t start t max X ln dx t+1 dx t=t t start Where t start exceeds the transient period.

Lyapunov Exponents: Numerical Estimation

function ls = lyapunov(f, F_deriv, n_samples, param_range, transient_time, max_time) % Takes a dynamical 1-parameter map and plots the Lyapunov exponents as a function of % the parameter % Create a vector of parameter values to evaluate param_values = linspace(param_range(1), param_range(2), n_samples); ls=[]; % This stores the Lyapunov exponents for param=param_values x_t=rand(1); % Sample over many initial values lyp_exps = []; for(t = 1:max_time) % Evaluate the user defined map F x_t=f(x_t,param); % Wait until the transient period is over if(t > transient_time) % Evaluate the derivative at the current point lyp_exps = [lyp_exps, F_deriv(x_t, param)]; end end end % Calculate Lyapunov approximation for the vector of derivatives ls = [ls, mean(log((abs(lyp_exps))))];

% Show the Lyapunov exponents and bifurcation plot for the logistic map F = @(x,r) r*x*(1-x) % Define the logistic map F_deriv =@(x,r) r-2*r*x % Define the derivative of the logistic map x0 = 0.5 % An initial value for the bifurcation plot n_samples = 250 % Number of points to plot param_range = [0,4] % Parameter range to plot, r for the logistic map transient_time = 500 % Make sure we are in the fixed point s basin max_time = 1000 % This minus the transient time is the number % of Lyapunov samples to average over % Calculate the Lyapunov exponents ls = lyapunov(f, F_deriv, n_samples, param_range, transient_time, max_time); % Calculate the long term populations bs = bifurcation(f,x0,param_range(1),param_range(2),n_samples,max_time); % Plot the results subplot(2,1,1) plot(linspace(param_range(1), param_range(2), n_samples), ls, b-'); subplot(2,1,2) plot(linspace(param_range(1), param_range(2), n_samples), bs, 'k.');

Top Panel: Plot of Lyapunov Exponents, Bottom Panel: Bifurcation Plot for r=0.5

Top Panel: Plot of Lyapunov Exponents, Bottom Panel: Bifurcation Plot for r=0.5

Lyapunov Exponents for 2D Maps The Du ng or Holmes Map: x t+1 = y t y t+1 = bx t + ay t y 3 t Chaotic at a =2.75 and b =0.2.

Lyapunov Exponents for 2D Maps This time we have to deal with divergence in 2-dimensions. To do that we use the Jacobian. See... an Lecture n-dimensional 5 for our discussion generalisation of the Jacobian. of f 0 (x)

function max_lyapunovs = lyapunov2d(f, F_Jacobian, t_max, param1_range, param2, x0, y0) current_l = 0; for param1=param1_range current_l = current_l + 1; % Initialize variables xy = [x0; y0]; xy_lengths = [1;0]; for i=1:t_max J = F_Jacobian(xy, param1, param2); xy=f(xy, param1, param2); end end end % Calculate divergence rate in the direction defined by the Jacobian xy_lengths=j*xy_lengths; length=sqrt(sum(xy_lengths.^2)); % Distance formula max_lyapunovs(current_l) = log(length)/i; % Calculate the average

F = @(xy,a, b) [xy(2); -b*xy(1)+a*xy(2)-xy(2)^3] % Duffling Map F_Jacobian = @(xy,a, b) [0 1; -b a-3*(xy(2))^2] % Duffling Jacobian max_time = 500; % How long to run (= number of samples to average) parameter1_range = 2:0.001:3; % Range over parameter 1: (a for Duffling Map) parameter2 = 0.2; % Fix parameter 2 (b for Duffling Map) % Initial values for x and y x0 = 0.5 y0 = 0.5 % Calculate the maximum Lyapunov exponents max_lyapunovs = lyapunov2d(f, F_Jacobian, max_time, parameter1_range, parameter2, x0, y0); % Make a plot of the maximum exponents with a line at 0 plot(parameter1_range,max_lyapunovs, parameter1_range, 0, 'k.') xlabel('param1: a', 'FontSize', 24); ylabel('\lambda_{max}', 'FontSize', 24);

Example: Holmes Map

R 3 (2 period attractor) R 3.449 (4 period) R 3.544 (8 period) R 3.564 (16 period)... R 3.569 ( period) Feigenbaum s Constant

Feigenbaum s Constant R 1 R 0 R 2 R 1 4.75 R n 1 R n 2 R n R n 1 4.669

Feigenbaum s Constant R 1 R 0 R 2 R 1 4.75 True for ALL maps that approach chaos by bifurcation. R n 1 R n 2 R n R n 1 4.669

Phase Space Contraction We can look at the evolution of a small volume (in 3D) as it contracts near an attractor The change in volume is measured by looking at the trace of the Jacobian, r ~ ~x

Dissipative vs Conserving Systems A dynamical system is dissipative, if it s phase space volume contracts continuously, ~r ~x(t) < 0, 8t

Dissipative vs Conserving Systems Dissapative or conservative? Gros, C., Complex and Adaptive Dynamical Systems, 4 th Edition

Dissipative vs Conserving Systems Conserving Dissapative Dissapative or conservative? Gros, C., Complex and Adaptive Dynamical Systems, 4 th Edition

Dissipative vs Conserving Systems A dynamical system is conserving, if it s phase space volume is constant, ~r ~x(t) =0, 8t

Adaptive Systems Adaptive systems are neither fully dissipative nor fully conserving. In terms of energy they have periods of taking up energy and periods of expending it. Technically, any system where ~ r ~x(t) can change sign is adaptive.

Outline Feigenbaum Constant (Order -> Chaos -> Deeper Order) Kuhns Revolutions in Science Microscope Telescope Computer (George Luger) May, Lorenz, Crutchfield, Mitchell and many others Biological Complexity from Simple Rules. Hearts (fractal structure) Exploring Dynamical Systems has demonstrated that surprising complexity can arise from simple rules. This allows us to understand the diversity generated by things like capitalism, biological evolution, computer architectures, and computer algorithms.